Multi-tiered quantization of channel state information in multiple antenna systems

BACKGROUND

One of the most promising solutions for increased spectral efficiency in high capacity wireless systems is the use of multiple antennas on fading channels. The fundamental issue in such systems is the availability of the channel state information (CSI) at transmitters and receivers. In general, if the receivers and transmitter have an access to CSI, the system throughput can be significantly increased. While it is usually assumed that perfect CSI is available at the receivers, the transmitter may only have partial CSI available due to the feedback delay and noise, channel estimation errors and limited feedback bandwidth, which forces CSI to be quantized at the receiver to minimize feedback rate. There is described here an improvement in the quantization of channel state information in a multiple antenna system.

SUMMARY

A multi-tiered CSI vector quantizer (VQ) is provided for time-correlated channels. The VQ operates for example by quantizing channel state information by reference to both current channel state information and a prior channel state quantization. A system is also provided that uses multi-tiered CSI quantizers. Enhanced signaling between the transmitter and receivers is provided in order to facilitate the use of multi-tiered CSI quantizers. These and other aspects of the device and method are set out in the claims, which are incorporated here by reference.

BRIEF DESCRIPTION OF THE FIGURES

Embodiments will now be described with reference to the figures, in which like reference characters denote like elements, by way of example, and in which:

<figref idrefs="DRAWINGS">FIG. 1</figref> is an illustration of a channel vector space according to known principles in the art;

<figref idrefs="DRAWINGS">FIG. 2</figref> is an illustration of the channel vector space of <figref idrefs="DRAWINGS">FIG. 1</figref> with a more fine-grained quantization;

<figref idrefs="DRAWINGS">FIG. 3</figref> is an illustration of the channel vector space of <figref idrefs="DRAWINGS">FIG. 2</figref> with a multi-tier quantization;

<figref idrefs="DRAWINGS">FIG. 4</figref> shows the structure of a quantization system;

<figref idrefs="DRAWINGS">FIG. 5</figref> shows the operation of an algorithm for designing a multi-tiered quantizer;

<figref idrefs="DRAWINGS">FIG. 6</figref> shows a region of one codebook and the corresponding regions of the corresponding next higher tier codebook for use in a multi-tiered quantizer;

<figref idrefs="DRAWINGS">FIG. 7</figref> shows an example of the operation of a multi-tiered quantizer;

<figref idrefs="DRAWINGS">FIG. 8</figref> is a flow diagram showing the operation of a multi-tiered quantizer;

<figref idrefs="DRAWINGS">FIG. 9</figref> shows the operation of the quantizer;

<figref idrefs="DRAWINGS">FIG. 10</figref> shows typical eigenmode and singular value coherence times; and

<figref idrefs="DRAWINGS">FIG. 11</figref> shows the results of a simulation comparing a multi-tiered quantizer with non-tiered quantizers.

DETAILED DESCRIPTION

In a multiple antenna system as for example shown in <figref idrefs="DRAWINGS">FIG. 4</figref>, information is transmitted over multiple channels <b>34</b> corresponding to multiple antennas <b>36</b>. Each channel <b>34</b> has a state that affects the propagation of information over the channel. The state of multiple channels <b>34</b> between a transmitter <b>32</b> and one or more receivers <b>30</b> in a multiple antenna system can be expressed as a vector. As the channel state changes, this vector moves through the channel vector space. The channel vector space may be separated into regions (see <figref idrefs="DRAWINGS">FIG. 1</figref>). Each region may be represented by an index. If each region corresponds to the part of the space closest, by some metric, to a particular member of a set of points in the space, then the regions are known as Voronoi regions <b>20</b> and the points are known as centroids <b>22</b>. In order to maximize throughput, it is preferred to associate each index to a centroid <b>22</b>, which represents the Voronoi region <b>20</b> which is the part of space closer to that centroid <b>22</b> than any other.

A vector quantizer (VQ) with multi-tiered quantization is aimed at transmission channels with memory, in which there is a need to reduce the feedback bandwidth and allow the system to automatically adjust the quantizer resolution to the rate of channel changes. In an exemplary design of a multi-tiered CSI vector quantizer (i.e., the description of centroids and Voronoi regions) for multiple-input, multiple-output (MIMO) channels with memory, the VQ uses multiple optimization steps.

In the typical CSI VQ, the quantization of the channel vector space can be illustrated as in <figref idrefs="DRAWINGS">FIG. 1</figref>: the CSI space is tessellated by Voronoi regions <b>20</b> with corresponding centroids <b>22</b> that represent all vector realizations within each Voronoi region (for each centroid, the corresponding Voronoi region is the set of points closer to that centroid than any other, according to some metric). The number of such regions (centroids) is defined by the number of available bits, which also influence the quantization error of the VQ. <figref idrefs="DRAWINGS">FIG. 1</figref> shows the situation when the channel CSI is correlated in time and follows some trajectory <b>28</b> in time.

The quantization error can be decreased if the CSI VQ resolution is increased using more bits in feedback link. <figref idrefs="DRAWINGS">FIG. 2</figref> shows the identical trajectory <b>28</b> of the channel vector realization as in <figref idrefs="DRAWINGS">FIG. 1</figref> with increased number of centroids <b>22</b> and Voronoi regions <b>20</b>A. In <figref idrefs="DRAWINGS">FIG. 1</figref> only two different centroid indices would be used to represent the channel trajectory, whereas in <figref idrefs="DRAWINGS">FIG. 2</figref>, four such indices would be used, which would result in more precise representation of the actual channel changes. The price for such improvement is a larger number of bits needed to characterize the quantized CSI indices that must be fed back to the transmitter.

The typical trajectory <b>28</b> of CSI vectors is partially predictable in a sense that the channel realizations between consecutive transmission epochs within the same frequency band are correlated. The correlation increases with decreasing relative speeds of receiver-transmitter pairs with the net effect of trajectories being statistically contained within a given Voronoi region for a predictable amount of time. The time metrics may be quantitative described in various ways such as using time metrics called eigemode coherence time and singular value coherence time. In CSI VQ context, the longer the coherence time, the less frequent the changes in VQ indices that need to be reported back to the transmitter.

A multi-tiered VQ allows for a significant reduction of the feedback rate for systems in which channel coherence times are fairly long. During the design of the quantizer, the Voronoi regions are optimized according to any chosen criterion in 2, 3, 4 and more tiers, in which consecutive Voronoi regions are embedded in the previous ones as shown for example in <figref idrefs="DRAWINGS">FIG. 3</figref> for a 2-tiered design.

In the example of <figref idrefs="DRAWINGS">FIG. 3</figref>, a 2-tiered CSI VQ is divided into primary <b>23</b> and secondary <b>24</b> Voronoi regions and corresponding centroids <b>25</b>, <b>26</b>. In the first phase of the VQ operation, only the primary regions are used to assign the primary centroid indices to the channel vectors. In the second phase, only the secondary centroids and the primary centroid within the first identified primary region are reported to the base station until the channel vector realization leaves the primary region. In this way, as long as the channel vector does not change very rapidly, high quantization resolution can be obtained at much lower feedback rate as the quantization points are concentrated within a space of single primary Voronoi region. Moreover, this mechanism allows the receiver to automatically adjust the vector resolution to the rate of channel changes. The transmitter and receiver must have a way of identifying for which Voronoi regions (primary, secondary etc.) the VQ indices are reported.

The following notation is used in describing an exemplary multi-tiered VQ: <ul><li id="ul0001-0001" num="0000"><ul><li id="ul0002-0001" num="0023">M—the number of tiers in the CSI vector quantizer design</li><li id="ul0002-0002" num="0024">m_k—the current tier index at receiver k</li><li id="ul0002-0003" num="0025">m^k—the current base station tier index of receiver k</li><li id="ul0002-0004" num="0026">N_m—the number of bits for CSI representation at each tier of the CSI MIMO VQ.</li></ul></li></ul>

A system using a dual VQ codebook design for quantization of channel state information in a multiple antenna system is shown in <figref idrefs="DRAWINGS">FIG. 4</figref>. This example shows a system according to the inventors' U.S. patent application Ser. No. 11/754,965 filed May 29, 2007. A multi-tiered VQ may be used for eigenmode and singular value codebooks in systems ranging from only one active receiver at a time to systems with multiple receivers being active simultaneously (where we define being active as receiving transmissions). The design of the multi-tiered codebooks can be applied to matrices of orthogonal eigenmodes, subsets of eigenmodes and scalar singular values as necessary. The following descriptions may be applied to any type of CSI quantizing solution.

In <figref idrefs="DRAWINGS">FIG. 4</figref>, a transmitter <b>32</b> communicates with a receiver <b>30</b> over a feedforward channel <b>66</b> and a feedback channel <b>38</b> using antennas <b>36</b>. The receiver <b>30</b> includes a channel estimator <b>40</b>, linear processor <b>68</b> for decoding a transmission, a singular value processing unit <b>42</b>, a power allocation and eigenmode selector <b>44</b>, and codebooks <b>48</b> and <b>46</b>. The receiver <b>30</b> may use various known electronic processors for its parts, and in one embodiment may use a monolithic application specific chip. The functions of the receiver <b>30</b> may be provided partly or entirely by hardware, firmware and/or software. The transmitter <b>32</b> includes an indexer and optimizer <b>54</b> and stored modulation and power allocation matrices <b>56</b> and <b>58</b> respectively. An input data stream <b>60</b> is fed to a modulator <b>62</b> that applies a linear modulation matrix selected from the stored modulation matrices <b>56</b>. The modulated data stream is fed to a power allocator <b>64</b>, which applies a power allocation matrix selected from the stored power allocation matrices <b>58</b>. The system of <figref idrefs="DRAWINGS">FIG. 4</figref> works as follows: <ul><li id="ul0003-0001" num="0000"><ul><li id="ul0004-0001" num="0029">1. Before the transmission epoch, each receiver <b>30</b> estimates <b>40</b> its channel matrix H <b>34</b> for the feedforward channel <b>66</b> and uses this information to perform <b>42</b> the singular value decomposition (SVD) of the matrix.</li><li id="ul0004-0002" num="0030">2. The eigenmode and singular value (power allocation) components are separately quantized <b>44</b> using two codebooks V <b>46</b> and D <b>48</b>, respectively.</li><li id="ul0004-0003" num="0031">3. The indices <b>50</b> of the selected codewords are fed back to the transmitter using a feedback channel <b>38</b>.</li><li id="ul0004-0004" num="0032">4. The transmitter uses all the indices from all receivers <b>50</b>, <b>52</b> in the system to choose <b>56</b>, <b>58</b> the pre-computed linear modulation and power allocation matrices B <b>62</b>, <b>64</b> and S, respectively. The choice is based on a predefined set of rules (maximum throughput, fairness, etc.),</li><li id="ul0004-0005" num="0033">5. The signal (x₁to x_N_{<sub2>T</sub2>}) <b>60</b> is modulated using the selected linear modulation and power allocation matrices B <b>62</b> and S <b>64</b> and transmitted via the feedforward channel <b>66</b>.</li><li id="ul0004-0006" num="0034">6. The transmitted modulated signal is processed by the receiver <b>68</b>. <br /> The transmitter <b>32</b> thus has a processor configured to carry out the above steps 4-5 and each receiver has one or more antennas <b>36</b> and a processor configured to carry out steps 1-3 and 6. </li></ul></li></ul>

Referring to <figref idrefs="DRAWINGS">FIG. 5</figref>, design of the multi-tiered codebooks (D or V) is performed as follows: <ul><li id="ul0005-0001" num="0000"><ul><li id="ul0006-0001" num="0036">1. Based on the desired system parameters (types of channels, required feedback rate required performance etc.) set parameters M and N_mfor each value of m=1, 2, . . . , M.</li><li id="ul0006-0002" num="0037">2. Set m=1 (step <b>70</b>).</li><li id="ul0006-0003" num="0038">3. Any of various vector quantizers may be used to design a receiver VQ using N₁resolution bits (step <b>72</b>). An example is given below, from U.S. patent application Ser. No. 11/754,965, which is entitled “Quantization of Channel State Information in Multiple Antenna Systems”.</li><li id="ul0006-0004" num="0039">4. Store the description of the Voronoi regions and centroids for m-tier of the VQ (step <b>74</b>).</li><li id="ul0006-0005" num="0040">5. If m is smaller than M (step <b>76</b>), continue the design in the following way: <ul><li id="ul0007-0001" num="0041">a) Create a large list of possible channel realizations (step <b>78</b>).</li><li id="ul0007-0002" num="0042">b) Using the m-tier VQ, quantize (step <b>80</b>) the above channel realizations.</li><li id="ul0007-0003" num="0043">c) Select channel realizations corresponding to each of the m-tier VQ indices (step <b>82</b>).</li><li id="ul0007-0004" num="0044">d) Within each of the m-tier Voronoi regions, perform m+1 tier design of a vector quantizer using any of various VQ designs, such as in U.S. patent application Ser. No. 11/754,965 and shown below. The algorithm uses N_m+1resolution bits within each region and forces one of the tier m+1 centroids within each region to be identical to the m-tier centroid corresponding to this region (step <b>84</b>). The codebook entries of the new codebook for each region are now considered to be the group of tier m+1 entries associated with the tier m codebook entry for that region.</li><li id="ul0007-0005" num="0045">e) The reused m-tier centroid is assigned N_m+1index bits equal to 0.</li><li id="ul0007-0006" num="0046">f) Increase m by 1 (step <b>86</b>).</li><li id="ul0007-0007" num="0047">g) Go to step 4.</li></ul></li><li id="ul0006-0006" num="0048">6. If m≧M, finish the VQ design (step <b>76</b>).</li><li id="ul0006-0007" num="0049">7. Design the modulation matrices corresponding to the designed multi-tiered VQ using any of various modulation matrix design techniques such as the algorithm shown in U.S. PTO application Ser. No. 11/754,965, as shown below (step <b>88</b>). The algorithm is now done (step <b>90</b>).</li></ul></li></ul>

The rationale behind re-using one of the m tier centroids at design phase of (m+1)-tier centroids and Voronoi regions (see bullet 5d above) is that the same set of modulation matrices can be used in a system where different users report their quantized channel information using different VQ tiers. As all m-tier centroids are contained in (m+1)-tier centroids, the effective indices can be easily used to decide which centroid must be used. <figref idrefs="DRAWINGS">FIG. 6</figref> shows an example: a single primary region with index 1111 92 and secondary regions with indices of 000, 001, etc 94 thus giving them effective indices of 1111000, 1111001, etc. 96 Note that the primary centroid is in the same place as the secondary centroid with index 000. Moreover, thanks to such a design, the base station may support users with different implementations of the vector quantizers, e.g., varying number of VQ tiers. Thanks to the embedding of the codewords, all such situations will be supported.

The algorithm from “Quantization of channel state information in multiple antenna systems” is as follows:

For the case of a single receiver active at a time, we introduce a heuristic distortion metric which is expressed as <br />γ_V(<i>n;H</i>)=∥<i>DV</i>^H<i>{circumflex over (V)}</i>(<i>n</i>)−<i>D∥</i>_F  (1)<br /> where {circumflex over (V)}(n) is the nth entry in the predefined set of channel diagonalization matrices and ∥·∥_Fis the Frobenius norm. We omitted subscript entries j in (1) for the clarity of presentation.

We assume that n=0, 1, . . . 2^N^{<sub2>V</sub2>}−1 where N_Vis the number of bits per channel realization in the feedback link needed to represent the vectors {circumflex over (V)}(n). To design the quantizer using (1), we divide the whole space of channel realizations H into 2^N^{<sub2>V </sub2>}regions V_iwhere <br /><i>V</i>_i<i>={H:γ</i>_V(<i>i;H</i>)&lt;γ_V(<i>j;H</i>)for all <i>j≠i}.</i>  (2)

The algorithm starts by creating a codebook of centroids {circumflex over (V)} and, based on these results, divides the quantization space into regions V_i. The codebook is created as follows: <ul><li id="ul0008-0001" num="0000"><ul><li id="ul0009-0001" num="0055">1. Create a large training set of L random matrices H(l).</li><li id="ul0009-0002" num="0056">2. For each random matrix H(l), perform singular value decomposition to obtain D(l) and V(l) as <br /><i>y</i>_j<i>=H</i>_j<i>x</i>_j<i>+n</i>_j=(<i>U</i>_j<i>D</i>_j<i>V</i>_j^H)(<i>V</i>_j<i>{tilde over (x)}</i>_j)+<i>n</i>_j  (3)</li><li id="ul0009-0003" num="0057">3. Set iteration counter i=0. Create a set of 2^N^{<sub2>V </sub2>}random matrices Ĥ(n).</li><li id="ul0009-0004" num="0058">4. For each matrix Ĥ(n) calculate corresponding {circumflex over (V)}⁽ⁱ⁾(n) using singular value decomposition.</li><li id="ul0009-0005" num="0059">5. For each training element H(l) and codebook entry {circumflex over (V)}⁽ⁱ⁾(n) calculate the metric in (1). For every l choose indexes n_opt(l) corresponding to the lowest values of γ_V(n; H(l)).</li><li id="ul0009-0006" num="0060">6. Calculate a new set {circumflex over (V)}⁽ⁱ⁺¹⁾(n) as a form of spherical average of all entries V(l) corresponding to the same index n using the following method. (The direct averaging is impossible since it does not preserve orthogonality between eigenvectors.) For all n calculate the subsets L(n)={l: n_opt(l)=n} and if their respective cardinalities |L(n)|≠0 the corresponding matrices <o>Q</o>⁽ⁱ⁺¹⁾(n) can be obtained as</li></ul></li></ul>

<maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msup><mover><mi>Q</mi><mi>_</mi></mover><mrow><mo>(</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></msup><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mo></mo><mrow><mi>L</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mo></mo></mrow></mfrac><mo>⁢</mo><mrow><munder><mo>∑</mo><mrow><mi>l</mi><mo>∈</mo><mrow><mi>L</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></mrow></munder><mo>⁢</mo><mrow><msup><mrow><mi>V</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mn>1</mn></msup><mo>⁢</mo><mi>O</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><msup><mrow><mi>V</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mi>H</mi></msup></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where ¹O is an n_T×n_Tall-zero matrix with the exception of the upper-left corner element equal to 1. Finally, using singular value decomposition, calculate {circumflex over (V)}⁽ⁱ⁺¹⁾(n) from <br /><i><o>Q</o></i>⁽ⁱ⁺¹⁾(<i>n</i>)=<i>{circumflex over (V)}</i>⁽ⁱ⁺¹⁾(<i>n</i>)<i>W</i>(<i>{circumflex over (V)}</i>⁽ⁱ⁺¹⁾(<i>n</i>))^H  (5)<br /> where W is a dummy variable. <ul><li id="ul0010-0001" num="0000"><ul><li id="ul0011-0001" num="0062">7. Calculate the average distortion metric</li></ul></li></ul>

<maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mrow><msubsup><mover><mi>γ</mi><mi>_</mi></mover><mi>V</mi><mrow><mo>(</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></msubsup><mo>=</mo><mrow><mrow><mn>1</mn><mo>/</mo><mi>L</mi></mrow><mo>⁢</mo><mrow><munder><mo>∑</mo><mi>l</mi></munder><mo>⁢</mo><mrow><mrow><msub><mi>γ</mi><mi>V</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>n</mi><mi>opt</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>;</mo><mrow><mi>H</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mrow></mrow></mrow></mrow></math></maths><ul><li id="ul0012-0001" num="0000"><ul><li id="ul0013-0001" num="0064">8. If distortion metric fulfills | <o>γ</o>_V⁽ⁱ⁺¹⁾− <o>γ</o>_V⁽ⁱ⁾|/ <o>γ</o>_V⁽ⁱ⁾&lt;⊖, stop. Otherwise increase i by 1 and go to 5).</li></ul></li></ul>

Upon completion of the above algorithm, the set of vectors {circumflex over (V)} can be used to calculate the regions in (2).

Having optimized power-independent entries in the codebook of channel eigenmode matrices {circumflex over (V)}, the next step is to create a codebook for power allocation Ŝ. We use a distortion metric defined as

<maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>γ</mi><mi>s</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>k</mi><mo>;</mo><mi>H</mi><mo>;</mo><mi>P</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mtable><mtr><mtd><mrow><mi>det</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>II</mi><mo>+</mo><mrow><mi>H</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>Q</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><msup><mi>H</mi><mi>H</mi></msup></mrow></mrow><mo>]</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi>det</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>II</mi><mo>+</mo><mrow><mi>H</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><mover><mi>V</mi><mo>^</mo></mover><mo>⁡</mo><mrow><mo>(</mo><msub><mi>n</mi><mi>opt</mi></msub><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mover><mi>S</mi><mo>^</mo></mover><mo>⁡</mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><msup><mover><mi>V</mi><mo>^</mo></mover><mi>H</mi></msup><mo>⁡</mo><mrow><mo>(</mo><msub><mi>n</mi><mi>opt</mi></msub><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>H</mi><mi>H</mi></msup></mrow></mrow><mo>]</mo></mrow></mrow></mtd></mtr></mtable></mrow></mtd><mtd><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where Ŝ(k) is the kth entry in the predefined set of channel water-filling matrices and {circumflex over (V)}(n_opt) is the entry in the {circumflex over (V)} codebook that minimizes metric (1) for the given H. We use k=0, 1 . . . 2^N^{<sub2>S</sub2>}−1 where N_Sis the number of bits per channel realization in the feedback link needed to represent the vectors Ŝ(k). Minimizing the metric in (6) is equivalent to minimizing the capacity loss between the optimum water-filling using Q and the quantized water-filling using {circumflex over (V)} and Ŝ.

Similarly to the previous problem, we divide the whole space of channel realizations H into 2^N^{<sub2>S </sub2>}regions S_i(P) where <br /><i>S</i>_i(<i>P</i>)={<i>H:γ</i>_S(<i>i;H;P</i>)&lt;γ_S(<i>j;H;P</i>)for all <i>j≠i}.</i>  (7)<br /> and to create the codebook Ŝ, we use the following method: <ul><li id="ul0014-0001" num="0000"><ul><li id="ul0015-0001" num="0069">1. Create a large training set of L random matrices H(l).</li><li id="ul0015-0002" num="0070">2. For each random matrix H(l), perform water-filling operation to obtain optimum covariance matrices Q(l) and S(l).</li><li id="ul0015-0003" num="0071">3. Set iteration counter i=0. Create a set of 2^N^{<sub2>S </sub2>}random diagonal matrices Ŝ^(s)(k) with Tr (Ŝ⁽ⁱ⁾(k))=P.</li><li id="ul0015-0004" num="0072">4. For ever codebook entry Ŝ⁽ⁱ⁾(k) and matrix Q(l), calculate the metric as in (6). Choose indexes k_opt(l) corresponding to the lowest values of γ_S(k; H(l); P).</li><li id="ul0015-0005" num="0073">5. If γ_S(k_opt(l); H(l); P)&gt;γ_eq(H(l); P) where γ_eq(H(l); P) is the metric corresponding to equal-power distribution defined as</li></ul></li></ul>

<maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><msub><mi>γ</mi><mi>eq</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>H</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>;</mo><mi>P</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mtable><mtr><mtd><mrow><mi>det</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>II</mi><mo>+</mo><mrow><mrow><mi>H</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mi>Q</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><msup><mi>H</mi><mi>H</mi></msup><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi>det</mi><mo>⁡</mo><mrow><mo>[</mo><mrow><mi>II</mi><mo>+</mo><mrow><mrow><mi>P</mi><mo>/</mo><msub><mi>n</mi><mi>T</mi></msub></mrow><mo>⁢</mo><mrow><mi>H</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><msup><mi>H</mi><mi>H</mi></msup><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>]</mo></mrow></mrow></mtd></mtr></mtable></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> set the corresponding entry k_opt(l)=2^N^{<sub2>S</sub2>}. For all k calculate the subsets L(k)={l: k_opt(l)=k}. <ul><li id="ul0016-0001" num="0000"><ul><li id="ul0017-0001" num="0075">6. For all k=0, 1, . . . 2^N^{<sub2>S</sub2>}−1 for which |L(k)|≠0, calculate a new set Ŝ⁽ⁱ⁺¹⁾(k) as the arithmetic average</li></ul></li></ul>

<maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msup><mover><mi>S</mi><mo>^</mo></mover><mrow><mo>(</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></msup><mo>⁡</mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mo></mo><mrow><mi>L</mi><mo>⁡</mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow><mo></mo></mrow></mtd></mtr></mtable><mo>⁢</mo><mrow><munder><mo>∑</mo><mrow><mi>l</mi><mo>∈</mo><mrow><mi>L</mi><mo>⁡</mo><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow></munder><mo>⁢</mo><mrow><mrow><mi>S</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>.</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><ul><li id="ul0018-0001" num="0000"><ul><li id="ul0019-0001" num="0077">7. Calculate the average distortion metric</li></ul></li></ul>

<maths id="MATH-US-00006" num="00006"><math overflow="scroll"><mtable><mtr><mtd><mrow><msubsup><mover><mi>γ</mi><mi>_</mi></mover><mi>s</mi><mrow><mo>(</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></msubsup><mo>=</mo><mrow><mfrac><mn>1</mn><mi>L</mi></mfrac><mo>⁢</mo><mrow><munder><mo>∑</mo><mi>l</mi></munder><mo>⁢</mo><mrow><mi>min</mi><mo>⁢</mo><mrow><mrow><mo>{</mo><mrow><mrow><msub><mi>γ</mi><mi>s</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><msub><mi>k</mi><mi>opt</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>;</mo><mrow><mi>H</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>;</mo><mi>P</mi></mrow><mo>)</mo></mrow></mrow><mo>,</mo><mrow><msub><mi>γ</mi><mi>eq</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>H</mi><mo>⁡</mo><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow><mo>;</mo><mi>P</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>}</mo></mrow><mo>.</mo></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>10</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><ul><li id="ul0020-0001" num="0000"><ul><li id="ul0021-0001" num="0079">8. If distortion metric fulfills | <o>γ</o>_S⁽ⁱ⁺¹⁾− <o>γ</o>_S⁽ⁱ⁾|/ <o>γ</o>_S⁽ⁱ⁾&lt;⊖, stop. Otherwise increase i with 1 and go to 4).</li></ul></li></ul>

The set of vectors Ŝ is then used to calculate the regions in (7). Since water-filling strongly depends on the power level P and {circumflex over (V)}, optimally the Ŝ should be created for every power level and number of bits N_Vin eigenvector matrix codebook (2).

In the multi-user case, we follow the approach of Spencer et al, where each user performs singular value decomposition of H_k=U_kS_kV_k^Hand converts its respective H_kto a n_T-dimensional vector h_kas <br />h_k=u_k^HH_k=s_k^maxv_k^H  (11)<br /> where s_k^maxis the largest singular value of S_kand u_kand v_kare its corresponding vectors from the unitary matrices U_kand V_k, respectively.

We use the linear block diagonalization approach, which eliminates MUI by composing the modulation matrix B [S] of properly chosen null-space eigenmodes for each set S. For each receiver i ε S, the ith row of the matrix H [S] is first deleted to form H [S_i]. In the next step, the singular value decomposition is performed to yield H [S_i]=U [S_i] S [S_i] V^H[S_i]. By setting the ith column of B[S] to be equal to the rightmost vector of V [S_i], we force the signal to the ith receiver to be transmitted in the null-space of the other users and no MUI will appear. In other words, the channel will be diagonalized with d_ibeing the entries on the diagonal of H [S] B [S]. This leads to formula

<maths id="MATH-US-00007" num="00007"><math overflow="scroll"><mtable><mtr><mtd><mrow><msup><mi>R</mi><mi>linear</mi></msup><mo>=</mo><mrow><munder><mi>max</mi><mi>S</mi></munder><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>n</mi><mi>T</mi></msub></munderover><mo>⁢</mo><msub><mrow><mo>[</mo><mrow><msub><mi>log</mi><mn>2</mn></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>ξ</mi><mo>⁡</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></mrow><mo>⁢</mo><msubsup><mi>d</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>+</mo></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>12</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where ξ [S] is the solution of the water-filling equation.

We assume that N_υ is the number of bits per channel realization in the feedback link needed to represent the vectors v_kin (11). We divide the space of all possible v's into 2^N^{<sub2>w </sub2>}regions υ_i<br />υ_i<i>={v:γ</i>_υ(<i>i;v</i>)&lt;γ_υ(<i>j;v</i>)for all <i>j≠i}</i>  (13)<br /> where γ_υ(n; v) is a distortion function. Within each region υ_i, we define a centroid vector {circumflex over (v)}(i), which will be used as a representation of the region. The design of the codebook {circumflex over (v)} can be done analytically and/or heuristically using for example the Lloyd algorithm. In this work, we define the distortion function as the angle between the actual vector v and {circumflex over (v)} (i): γ_υ(i; v)=cos⁻¹({circumflex over (v)}(i)·v), which has been shown by Roh and Rhao to maximize ergodic capacity, and use Lloyd algorithm to train the vector quantizer. Note that the construction of {circumflex over (v)} is independent of the transmit power.

We assume that N_sis the number of bits per channel realization in the feedback link needed to represent the scalar s_k^maxin (11). We divide the space of all possible channel realizations s=s^maxinto 2^N, regions s_i<br /><i>s</i>_i<i>={s:|ŝ</i>(<i>i</i>)−<i>s|&lt;|ŝ</i>(<i>j</i>)−<i>s</i>| for all <i>j≠i}</i>  (14)<br /> where ŝ(i) are scalar centroids representing regions s_i. In this work, we perform the design of the codebook ŝ using the classical non-uniform quantizer design algorithm with distortion function given by quadratic function of the quantization error as ε(i; s)=(s−ŝ(i))².

The construction of the codebook ŝ is generally dependent on the transmit power level. However, the differences between the codebooks ŝ for different power regions are quite small. This allows us to create only one codebook ŝ and use it for all transmit powers.

The calculation of the modulation matrix {circumflex over (B)} is based on the given codebook {circumflex over (v)}. We assume that the quantization of the channel eigenmodes is performed at the receiver side and each user transmits back its codebook index i_k. The indices are then used at the transmitter side to select the modulation matrix {circumflex over (B)}(i₁, i₂, . . . i_k). Since, from the linear transmitter point of view, ordering of the users is not important, we will use the convention that the indices (i₁, i₂, . . . i_K) are always presented in the ascending order. For example, in a system with K=2, n_T=2 and 1-bit vector quantizers {circumflex over (v)}, there will exist only three possible modulation matrices corresponding to sets of {circumflex over (v)} indices (1, 1) (1, 2) and (2, 2).

In the context of vector quantizing, the design of the modulation matrices can no longer be based on the algorithm presented for the single user case. Using this method with quantized versions of h_kproduces wrong result when identical indices i_kare returned and the receiver attempts to jointly optimize transmission to the users with seemingly identical channel vectors ĥ_k. Instead, we propose the following algorithm to optimize the set of matrices {circumflex over (B)}(i₁, i₂, . . . i_K): <ul><li id="ul0022-0001" num="0000"><ul><li id="ul0023-0001" num="0089">1. Create a large set of Nn_Trandom matrices H_k, where N is the number of training sets with n_Tusers each.</li><li id="ul0023-0002" num="0090">2. For each random matrix H_k, perform singular value decomposition and obtain h_kas in (11).</li><li id="ul0023-0003" num="0091">3. For each vector h_kstore the index i_kof the corresponding entry {circumflex over (v)}(i_k).</li><li id="ul0023-0004" num="0092">4. Divide the entire set of matrices H_kinto N sets with n_Telements each.</li><li id="ul0023-0005" num="0093">5. Sort the indices i_kwithin each set l in the ascending order. Map all unique sets of sorted indices to a set unique indices I_B(for example (1, 1)→I_B=1; (1, 2)→I_B=2; (2, 2)→I_B=3 . . . ).</li><li id="ul0023-0006" num="0094">6. In each set l, reorder the corresponding channel vectors h_kaccording to their indices i_kand calculate the B_lusing the block diagonalization method described above.</li><li id="ul0023-0007" num="0095">7. Calculate a set {circumflex over (B)}(I_B) as a column-wise spherical average of all entries B_lcorresponding to the same index I_B.</li></ul></li></ul>

After calculation of |I_B| modulation matrices {circumflex over (B)}, the remaining part of system design is the calculation of the water-filling matrices {circumflex over (D)}, which divide the powers between the eigenmodes at the transmitter. The procedure for creation of codebook {circumflex over (D)} is similar to the above algorithm, with the difference that the entries ŝ(n_k) are used instead of {circumflex over (v)}(i_k), and the spherical averaging of the water-filling matrices is performed diagonally, not column-wise. Explicitly: <ul><li id="ul0024-0001" num="0000"><ul><li id="ul0025-0001" num="0097">1. Create a large set of Nn_Trandom matrices H_k, where N is the number of training sets with n_Tusers each.</li><li id="ul0025-0002" num="0098">2. For each random matrix H_k, perform singular value decomposition and obtain h_kas in (11).</li><li id="ul0025-0003" num="0099">3. For each vector h_kstore the index n_kof the corresponding entry ŝ(n_k).</li><li id="ul0025-0004" num="0100">4. Divide the entire set of matrices H_kinto N sets with n_Telements each.</li><li id="ul0025-0005" num="0101">5. Sort the indices n_kwithin each set l in the ascending order. Map all unique sets of sorted indices to a set of unique indices I_D(for example (1, 1)→I_D=1; (1, 2)→I_D=2; (2, 2)→I_D=3 . . . ).</li><li id="ul0025-0006" num="0102">6. In each set l, reorder the corresponding channel vectors h_kaccording to their indices n_kand calculate the optimum D_iusing the method of waterfilling of (12).</li><li id="ul0025-0007" num="0103">7. Calculate a set {circumflex over (D)}(I_D) as a diagonal spherical average of all entries D_icorresponding to the same index I_D.</li></ul></li></ul>

Referring to <figref idrefs="DRAWINGS">FIGS. 7 and 8</figref>, based on the design of the multi-tiered codebooks D and V as in the previous section, the system will operate as follows: <ul><li id="ul0026-0001" num="0000"><ul><li id="ul0027-0001" num="0105">1. Initialize transmission epoch to t=1.</li><li id="ul0027-0002" num="0106">2. Set m_k=1 at each receiver k, (all users will use separate indices m_k) (step <b>112</b>).</li><li id="ul0027-0003" num="0107">3. Set m^k=1 separately for each receiver at the transmitter side. The transmitter-side indices m^kshould be mapped to their respective receiver-side indices m_k(step <b>110</b>).</li><li id="ul0027-0004" num="0108">4. Each receiver estimates its channel matrix H[t] (step <b>40</b>).</li><li id="ul0027-0005" num="0109">5. Each receiver performs the vector quantization of the channel using m=1 tier quantizers described above (step <b>114</b>).</li><li id="ul0027-0006" num="0110">6. The m-tier N_m-bit long indices are fed back to the transmitter.</li><li id="ul0027-0007" num="0111">7. The transmitter performs the selection of active users using any method (maximum fairness, maximum throughput etc.) and chooses the optimum modulation matrices using a VQ method such as the method described in U.S. patent application Ser. No. 11/754,965.</li><li id="ul0027-0008" num="0112">8. The signal is transmitted to the selected active receivers.</li><li id="ul0027-0009" num="0113">9. Increase transmission epoch as t=t+1.</li><li id="ul0027-0010" num="0114">10. Each receiver estimates its channel matrix H[t] (step <b>40</b>).</li><li id="ul0027-0011" num="0115">11. Each receiver performs the vector quantization of the channel using m-tier quantizers described above (step <b>114</b>).</li><li id="ul0027-0012" num="0116">12. Each receiver that recognizes (step <b>116</b>) that its quantized channel's m_k-tier Voronoi region in the t+1 epoch is identical to the m_k-tier Voronoi region in epoch t performs the following steps: <ul><li id="ul0028-0001" num="0117">a) Unless 118 m_k=M, increase the receiver's index to m_k=m_k+1 (step <b>120</b>).</li><li id="ul0028-0002" num="0118">b) The channel realization within the unchanged Voronoi region is quantized using the new m_k-tier quantizer (step <b>114</b>).</li><li id="ul0028-0003" num="0119">c) The receiver uses a known mechanism (see later in the document) to signal to the transmitter the new m_k-tier of VQ.</li><li id="ul0028-0004" num="0120">d) The N_mkbits long indices are fed back to the transmitter (step <b>124</b>).</li><li id="ul0028-0005" num="0121">e) Transmitter increases its index as m^k=m^k+1 (step <b>126</b>).</li></ul></li><li id="ul0027-0013" num="0122">13. Each receiver that recognizes (step <b>116</b>) that its quantized channel's m_k-tier Voronoi region in the t+1 epoch is not identical to the m_k-tier Voronoi region in epoch t performs the following steps: <ul><li id="ul0029-0001" num="0123">a) Unless (step <b>128</b>) m_k=1, decrease the receiver's index m_kto the last tier where the Voronoi regions m_k−1are identical in both t and t+1 epochs (step <b>130</b>).</li><li id="ul0029-0002" num="0124">b) If no such tier can be found, set m_k=1, otherwise update the m_kto a value for which Voronoi regions m_k−1are identical (step <b>130</b>).</li><li id="ul0029-0003" num="0125">c) The channel realization is quantized using the m_k-tier quantizer (step <b>114</b>).</li><li id="ul0029-0004" num="0126">d) The receiver uses a known mechanism (see below) to signal to the transmitter the new m_ktier of VQ.</li><li id="ul0029-0005" num="0127">e) The N_mkbits long indices are fed back to the transmitter (step <b>124</b>).</li><li id="ul0029-0006" num="0128">f) Transmitter decreases its index as m^k=m_k(step <b>134</b>).</li></ul></li><li id="ul0027-0014" num="0129">14. The transmitter selects the modulation matrices based on the indices fed from the receivers and each receiver's separate m^kindex stored at the transmitter side (step <b>136</b>).</li><li id="ul0027-0015" num="0130">15. The modulation matrices are used to transmit the information to the selected receivers (step <b>138</b>). <br /> The example of the algorithm's operation shown in <figref idrefs="DRAWINGS">FIGS. 7-8</figref> for one mobile receiver uses M=3 tiered quantizer. In this scenario, CSI vector stays in tier-1 Voronoi region <b>20</b> in first 5 frames F<b>1</b>-F<b>5</b>, in the tier-2 region <b>24</b> in first 5 frames F<b>1</b>-F<b>5</b>, and tier-3 region <b>100</b> in frames F<b>2</b> and F<b>3</b>. Receiver R<b>1</b> recognizes the subsequent Voronoi regions and adjusts the tier (m) <b>104</b> of the used quantizer accordingly by increasing and decreasing quantizer resolution. The quantizer indices each representing the centroid <b>22</b>,<b>26</b>,<b>102</b> of its respective Voronoi region in the appropriate tier, are then fed to the base station B<b>1</b> that combines them properly so that the effective CSI resolution N varies in time (t increases from F<b>1</b>-F<b>6</b>) depending on the rate of channel changes. Base station B<b>1</b> chooses modulation matrix <b>56</b> and transmits signal <b>60</b>. At each time the resolution is equal to that of an untiered quantizer with a number of bits equal to the number of bits <b>106</b> representing the tier used (N_m) plus that for all lower tiers. It can be clearly seen that the proposed algorithm allows the system to automatically adjust the resolution to the speed of channel changes. </li></ul></li></ul>

<figref idrefs="DRAWINGS">FIG. 9</figref> shows a graphical representation of the algorithm.

The set of indices m_kat the receivers should be matched to the indices m_kat the transmitter. If the transmitter uses the index m^kthat corresponds to the wrong m_k-tier of the receiver VQ, the resulting loss of performance may be very significant. In general, the index of the quantized channel vector at the transmitter is reconstructed as: <ul><li id="ul0030-0001" num="0133">AAAABBBCCC . . . <br /> where AAAA corresponds to m=1 tier N₁indexing bits, BBB corresponds to m=2 tier N₂indexing bits etc. (see <figref idrefs="DRAWINGS">FIG. 6</figref> for an example). </li></ul>

At any given time, the transmitter receives only m-tier index bits (AAAA, BBB, CCC etc.) and it must be able to establish which tier those bits correspond to. For example, there must be a signaling method allowing the transmitter to distinguish between two consecutive transmissions such as BBB, BBB where the channel vector moved away from one tier-2 centroid to another, from the BBB, CCC transmission, where the channel vector stayed in the same tier-2 region BBB and tier-3 quantization was used in the CCC word.

Various methods may be used for transmitting index information from the receiver to the transmitter such as: <ul><li id="ul0031-0001" num="0000"><ul><li id="ul0032-0001" num="0136">1. Direct indexing of VQ words. In order to let the transmitter know, which VQ tier is used, the actual VQ codeword index is extended with the bit representation of the index m_kof each receiver. Example:</li><li id="ul0032-0002" num="0137">MMAAAA, MMBBB, MMCCC . . .</li><li id="ul0032-0003" num="0138">Where, for example, two bits MM are used to represent one of the four quantization tiers in the system. The drawback of this method is that the additional feedback load is required to transmit information about m_kindices.</li><li id="ul0032-0004" num="0139">2. Varying length of different m tier VQ words. In this method, each m-tier of the CSI quantizer is characterized by different number of indexing bits N_m. Example:</li><li id="ul0032-0005" num="0140">AAAA, BBB, CC, D . . .</li><li id="ul0032-0006" num="0141">where four bits AAA are used to represent tier-1 quantization, 3 bits BBB are used to represent tier-2 quantization etc. The advantage of this system is that there is no need to transmit additional bits M as in the previous method. The drawback of this method is that there must be another mechanism allowing the transmitter to count how many bits were actually sent from the receiver and the varying feedback load.</li><li id="ul0032-0007" num="0142">3. Channel prediction based assessment of VQ tier. In this method, each m-tier of the CSI quantizer may be characterized by any number of bits N_mand the statistical channel characterization is used by the transmitter to decide whether the channel vector stayed in the previous m-tier Voronoi region or moved away from it. The advantage of this system is that it leaves a large degree of freedom for designing the feedback link. The drawback of this method is that the complexity of transmitter design grows and there may be erroneous decisions on the tier of quantizer used by the receivers.</li><li id="ul0032-0008" num="0143">4. Hybrid solutions combining the previous three methods in any way that is suitable from system design point of view.</li></ul></li></ul>

In the course of the system operation, it may happen that some of the transmitter indices m^kwill no longer be synchronized with corresponding receiver indices m_k. Such a situation will typically happen when one of the feedback messages from a receiver has not been detected at the transmitter (i.e., the transmitter lacks channel quantization index for the current transmission epoch) or the received message with the indexing information does not agree with the expected quantization tier m.

In practical communication systems, two erroneous situations can occur: <ul><li id="ul0033-0001" num="0000"><ul><li id="ul0034-0001" num="0146">The received feedback message shows the situation when m_k&gt;m^k+1. Such a situation is not allowed during the course of the normal operation since the receiver may only step back to the lower tier quantizers or increase the current one by 1.</li><li id="ul0034-0002" num="0147">The transmitter did not receive any feedback information due to the feedback link problems.</li></ul></li></ul>

Various methods may be used to solve the problem such as: <ul><li id="ul0035-0001" num="0000"><ul><li id="ul0036-0001" num="0149">1. Use channel prediction to recover the incorrect index. The previously used indices are used to extrapolate the actual channel information index.</li><li id="ul0036-0002" num="0150">2. Deactivate the user for the next transmission epoch and send the VQ RESET message to it. If the transmitter cannot reliably decide, which channel index was reported by the receiver, it sends a special VQ RESET message to the receiver containing the last value of the effective index at the transmitter AAAABBBCC . . . without the last tier of bits (in other words, bits up to the level m^k−1 are communicated to the base station). The receiver than establishes, whether the same tier of the VQ can be used or whether it has to step back to a lower tier. The new indices are sent to the transmitter and the system resumes the usual operation.</li></ul></li></ul>

<figref idrefs="DRAWINGS">FIG. 10</figref> shows a typical set of curves representing the eigenmode and singular value coherence times for a 2×2 MIMO system. For definitions see B. Mielczarek and W. Krzymien, “Influence of CSI feedback delay on capacity of linear multi-user MIMO systems,” in Proc. IEEE WCNC., Hong Kong, March 2007, pp. 1188-1192. As one can see, the length of time in which the first tier Voronoi regions do not change decreases with increasing resolution and normalized Doppler frequency of the channel f_DT_frame. For example, with f_DT_frame=0.02, the Voronoi regions of the eigenmode quantizers will stay the same for approximately 6 consecutive frames when the eigenmode quantizer uses N=4 bit resolution. When the quantizer uses N=7 bits, only 2-3 consecutive frames will have identical first tier Voronoi regions. In general, in order to improve system's throughput, it is preferable to use higher resolution of VQ but the price for the improvement is the high feedback burden and frequent changes of the high resolution indices. As shown in example below, by using a multi-tier VQ design, it is actually possible to achieve very good system performance and significantly reduce the required feedback bit rate.

In <figref idrefs="DRAWINGS">FIG. 11</figref>, the simulation results are shown for a system with 2-tier eigenmode quantizer (M=2) using N₁=4 and N₂=3. The system has been designed using algorithms described in this application and starts by transmitting 4 bits (AAAA) and, if the 1-tier Voronoi regions are the same in the consecutive frames, only 3 bits (BBB) are sent to the transmitter. We compare them with 1-tier systems with N₁=7 and N₁=4. The implemented algorithms are tested on a system with 2 base station antennas and 10 users with 2 receive antennas each. We test three channels with maximum normalized Doppler frequencies equal to 0.01, 0.02 and 0.1 as in <figref idrefs="DRAWINGS">FIG. 11</figref>. As one can see, the throughput gap between the conventional 1-tier vector quantizers for N₁=7 and N₁=4 is quite large (around 3 dB at 10 bpcu)—any increase of throughput in such simple systems requires increase of feedback bandwidth. However, if the channel is assumed to have memory, by using our proposed approach, it is possible to attain almost the same performance with multi-tier CSI quantization. In our example, the maximum feedback burden is set to 4 bits/frame/receiver for the 2-tier system but the performance is almost equivalent to 7 bit feedback system for a large range of Doppler frequencies. Hence, by proper choice of the number of tiers and their corresponding resolutions, it is possible to design the practical systems for a wide variety of channel conditions, required throughput performance and maximum feedback link bit rates.

In the claims, the word “comprising” is used in its inclusive sense and does not exclude other elements being present. The indefinite article “a” before a claim feature does not exclude more than one of the feature being present. Each one of the individual features described here may be used in one or more embodiments and is not, by virtue only of being described here, to be construed as essential to all embodiments as defined by the claims.

Immaterial modifications may be made to the embodiments described here without departing from what is covered by the claims.